PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

DTFT, DFT, ZT, and FFT 525

Example 5.9

Verify the linearity property of DFT that states, in short, the following:

The sum of DFT[f 1 (n)] + DFT[f 2 (n)] + DFT[f 3 (n)] ... is equal to the DFT of the sums of

(f 1 (n) + f 2 (n) + f 3 (n) ...), for the following specifi c case:

DFT u n DFT u n DFT

n

un

n

[()]() cos ()

1

3





































^73

 





































DFT u nu n

n

un n

n

() () cos ()

1

3

7 1

3

for



            16

where

f 1 (n) = u(n)

fn un

n

2 ()
 ()

1

3









fn

n

3 ()
^7 cos un()

3

 







FIGURE 5.38

Plots of DFT of signal(t) = noise + f(t) of Example 5.8.

0

− 4

− 2

0

Angle

2

4

500 1000 1500 2000

frequency in Hertz

2500 3000 3500 4000

0

0

50

100

Magnitude

150

500 1000 1500 2000

DFT Magnitude and Phase

frequency in Hertz

2500 3000 3500 4000